Mathematic constraints as cause of emergent properties

[KALSTER]

This could be difficult and tedious to follow as a result of the lack of proper terms on my part , so please bare with me. I’d like to know if it is possible for a mathematical constraint to be the cause of emergent properties or self organisation.

 

(1)Lets say that you have an infinite number of points on a boundless 2D plane. The distance between any two points is a 1D line/string, and is related by a probability curve. This curve is , where x > 0; x is Distance and y is Probability (The first quadrant half of a Hyperbola).

 

It is my understanding that there can be structure within infinity. An example of this is that in a hypothetical, infinite and unbounded universe, it would contain an infinite amount of matter, whether it consists of one proton every light year or if it is one continuous gas cloud. Similarly I am thinking that such an arrangement can exhibit structure as well, that is, areas of varying density.

 

I am not sure how exactly to frase this. :?

 

OK. If you start with any given physical system, you can analise the workings of such a system by observing behaviour and then trying to find the causes behind each occurrence. Eventually one would start to see patterns emerging, patterns that could be described by equations/formulas. Each pattern could be further analysed until the cause and effect relationships between constituent participants in the pattern can be deduced. This process can be repeated again and again, further reducing the system to a larger number of constituent predictable processes each time, but then eventually a limit is reached, which can be the limit of computing power, etc. I am wondering, after sufficiently reducing the system, if one could eventually reach a point where a simple mathematic expression can be the direct cause of all the macro observed effects.

 

Take the setup at (1). The distance between any two points tend toward 0.

If the formula is valid for an infinite plane, could a clumping of points, that is nearer to each other than surrounding points, actually directly cause the surrounding points to be further apart from each other in order for the formula to stay exactly valid? That is, if a clump of higher density points are formed locally, that the “violation” of the formula could cause an equal amount of deviation in the opposite direction (lower density) in the surrounding point space, starting from a maximum deviation at the clump’s border and petering out to zero deviation.

 

Would this require base/minimum units of distance and/or time to be possible?

How (if at all) would other areas of higher average density be affected by areas of lower density it might be passing into? Would further parameters be required for one high density area to affect another (by way of the low density “aura”)?